Chaotic Dynamics of Cantilever Beams with Breathing Cracks

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DOI https://doi.org/10.15407/pmach2025.01.033
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute of Power Machines and Systems
of National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 28, no. 1, 2025 (March)
Pages 33-41
Cited by J. of Mech. Eng., 2025, vol. 28, no. 1, pp. 33-41

 

Authors

Serhii Ye. Malyshev, National Technical University “Kharkiv Polytechnic Institute” (2, Kyrpychova str., Kharkiv, 61002, Ukraine), Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), ORCID: 0009-0000-7739-9230

Kostiantyn V. Avramov, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61046, Ukraine), e-mail: kvavramov@gmail.com, ORCID: 0000-0002-8740-693X

 

Abstract

A nonlinear dynamic system with a finite number of degrees of freedom, which describes the forced oscillations of a beam with two breathing cracks, is obtained. The cracks are located on opposite sides of the beam. The Bubnov-Galerkin method is used to derive the nonlinear dynamic system. Infinite sequences of period-doubling bifurcations cause chaotic oscillations and are observed at the second-order subharmonic resonance. Poincaré sections and spectral densities are calculated to analyze the properties of chaotic oscillations. In addition, Lyapunov exponents are calculated to confirm the chaotic behavior. As follows from the numerical analysis, chaotic oscillations arise as a result of the nonlinear interaction between cracks.

 

Keywords: cracked beam, forced oscillations, period-doubling bifurcation, chaotic oscillations, Lyapunov exponent.

 

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Received 25 November 2024

Accepted 20 December 2024

Published 30 March 2025